11–14. Working with parametric equations Consider the following parametric equations.
a. Make a brief table of values of t, x, and y.
b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).
x=−t+6, y=3t−3; −5≤t≤5
11–14. Working with parametric equations Consider the following parametric equations.
a. Make a brief table of values of t, x, and y.
b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).
x=2 t,y=3t−4;−10≤d≤10
The ellipse and the parabola: Let R be the region bounded by the upper half of the ellipse x²/2 + y² = 1 and the parabola y = x²/√2
a. Find the area of R
Reflection property of parabolas: Consider the parabola y = x²/(4p) with its focus at F(0, p). The goal is to show that the angle of incidence (α) equals the angle of reflection (β).
a. Let P(x₀, y₀) be a point on the parabola. Show that the slope of the tangent line at P is tan θ = x₀/(2p).
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The point with Cartesian coordinates (−2, 2) has polar coordinates (2√2, 3π/4), (2√2, 11π/4), (2√2, −5π/4), and (−2√2,−π/4).
67–72. Derivatives Consider the following parametric curves.
a. Determine dy/dx in terms of t and evaluate it at the given value of t.
x = t + 1/t, y = t − 1/t; t = 1
